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  • Identical expressions

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  • (xln(x)sin(x))/(x6-1)
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  • (xln(x)sin(x)) divide by (x^6-1)

  • Similar expressions

  • (xln(x)sin(x))/(x^6+1)
  • (xln(x)sinx)/(x^6-1)
Sum of series/ (xln(x)sin(x))/(x^6-1)

Sum of series (xln(x)sin(x))/(x^6-1)


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The solution

You have entered [src] 
  oo                 
____                 
\   `                
 \    x*log(x)*sin(x)
  \   ---------------
  /         6        
 /         x  - 1    
/___,                
n = 1
∑n=1∞xlog⁡(x)sin⁡(x)x6−1\sum_{n=1}^{\infty} \frac{x \log{\left(x \right)} \sin{\left(x \right)}}{x^{6} - 1} 
Sum(((x*log(x))*sin(x))/(x^6 - 1), (n, 1, oo))
The radius of convergence of the power series
Given number:
xlog⁡(x)sin⁡(x)x6−1\frac{x \log{\left(x \right)} \sin{\left(x \right)}}{x^{6} - 1}
It is a series of species
an(cx−x0)dna_{n} \left(c x - x_{0}\right)^{d n}
- power series.
The radius of convergence of a power series can be calculated by the formula:
Rd=x0+lim⁡n→∞∣anan+1∣cR^{d} = \frac{x_{0} + \lim_{n \to \infty} \left|{\frac{a_{n}}{a_{n + 1}}}\right|}{c}
In this case
an=xlog⁡(x)sin⁡(x)x6−1a_{n} = \frac{x \log{\left(x \right)} \sin{\left(x \right)}}{x^{6} - 1}
and
x0=0x_{0} = 0
,
d=0d = 0
,
c=1c = 1
then
1=lim⁡n→∞11 = \lim_{n \to \infty} 1
Let's take the limit
we find
True

False
The answer [src] 
oo*x*log(x)*sin(x)
------------------
           6      
     -1 + x
∞xlog⁡(x)sin⁡(x)x6−1\frac{\infty x \log{\left(x \right)} \sin{\left(x \right)}}{x^{6} - 1} 
oo*x*log(x)*sin(x)/(-1 + x^6)

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